Discussion of Homotopy Type Theory and Univalent Foundations
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From: "Jon Sterling" <jon@jonmsterling.com>
To: "steve awodey" <steveawodey@icloud.com>
Cc: "'Martin Escardo' via Homotopy Type Theory"
	<HomotopyTypeTheory@googlegroups.com>
Subject: Re: [HoTT] Re: A question about the problem with regularity in CCHM cubical type theory
Date: Sun, 15 Sep 2019 21:44:38 -0400
Message-ID: <afe67df2-7f4a-4e51-abc9-3f8e3a985623@www.fastmail.com> (raw)
In-Reply-To: <A605E6EE-0101-4390-B50D-A6AEB36FDCC2@icloud.com>

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Thanks Steve!

On Sun, Sep 15, 2019, at 9:38 PM, steve awodey wrote:
> It does: the relf term is always a weak equivalence by 3 for 2, and it’s monic because it’s a section. 
> 
> Sent from my iPhone
> 
> > On Sep 15, 2019, at 21:04, Jon Sterling <jon@jonmsterling.com> wrote:
> > 
> > Hi Andrew,
> > 
> > Does "all monomorphisms are cofibrations" imply that identity and path types coincide? or only the other way around?
> > 
> > Thanks,
> > Jon
> > 
> > 
> >> On Sun, Sep 15, 2019, at 7:55 AM, Andrew Swan wrote:
> >> You might have already seen this, but I have a paper on some related 
> >> issues at https://arxiv.org/abs/1808.00920 . In that paper I didn't 
> >> look at the original version of regularity, but a more recent version 
> >> ("all monomorphisms are cofibrations") that fits better with the 
> >> general framework of Orton and Pitts. In that case it is definitely 
> >> equality of objects that causes problems.
> >> 
> >> Best,
> >> Andrew
> >> 
> >>> On Friday, 13 September 2019 08:10:42 UTC+2, Jasper Hugunin wrote:
> >>> Hello all,
> >>> 
> >>> I've been trying to understand better why composition for the universe does not satisfy regularity.
> >>> Since comp^i [ phi |-> E ] A is defined as (roughly) Glue [ phi |-> equiv^i E ] A, I would expect regularity to follow from two parts:
> >>> 1. That Glue [ phi |-> equivRefl A ] A reduces to A (a sort of regularity condition for the Glue type constructor itself)
> >>> 2. That equiv^i (refl A) reduces to equivRefl A
> >>> I'm curious as to which (or both) of these parts was the issue, or if regularity for the universe was supposed to follow from a different argument.
> >>> 
> >>> Context:
> >>> I've been studying and using CCHM cubical type theory recently, and often finding myself wishing that J computed strictly.
> >>> If I understand correctly, early implementations of ctt did have strict J for Path types, and this was justified by a "regularity" condition on the composition operation, but as discussed in this thread on the HoTT mailing list <https://groups.google.com/d/msg/homotopytypetheory/oXQe5u_Mmtk/3HEDk5g5uq4J>, the definition of composition for the universe was found to not satisfy regularity.
> >>> I don't remember seeing the regularity condition defined anywhere, but my understanding is that it requires that composition in a degenerate line of types, with the system of constraints giving the sides of the box also degenerate in that direction, reduces to just the bottom of the box. This seems to be closed under the usual type formers, plus Glue, but not the universe with computation defined as in the CCHM paper <http://www.cse.chalmers.se/~coquand/cubicaltt.pdf> (for trivial reasons and non-trivial reasons; it gets stuck at the start with Glue [ phi |-> equiv^i refl ] A not reducing to anything).
> >>> 
> >>> Best regards,
> >>> - Jasper Hugunin
> >> 
> >> -- 
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> >> To view this discussion on the web visit 
> >> https://groups.google.com/d/msgid/HomotopyTypeTheory/16b3b92f-069b-468c-94f8-f3859e152338%40googlegroups.com <https://groups.google.com/d/msgid/HomotopyTypeTheory/16b3b92f-069b-468c-94f8-f3859e152338%40googlegroups.com?utm_medium=email&utm_source=footer>.
> > 
> > -- 
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> 
> 

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<!DOCTYPE html><html><head><title></title><style type="text/css">p.MsoNormal,p.MsoNoSpacing{margin:0}</style></head><body><div style="font-family:Arial;">Thanks Steve!</div><div style="font-family:Arial;"><br></div><div>On Sun, Sep 15, 2019, at 9:38 PM, steve awodey wrote:<br></div><blockquote type="cite" id="qt"><div style="font-family:Arial;">It does: the relf term is always a weak equivalence by 3 for 2, and it’s monic because it’s a section.&nbsp;<br></div><div style="font-family:Arial;"><br></div><div style="font-family:Arial;">Sent from my iPhone<br></div><div style="font-family:Arial;"><br></div><div style="font-family:Arial;">&gt; On Sep 15, 2019, at 21:04, Jon Sterling &lt;jon@jonmsterling.com&gt; wrote:<br></div><div style="font-family:Arial;">&gt;&nbsp;<br></div><div style="font-family:Arial;">&gt; Hi Andrew,<br></div><div style="font-family:Arial;">&gt;&nbsp;<br></div><div style="font-family:Arial;">&gt; Does "all monomorphisms are cofibrations" imply that identity and path types coincide? or only the other way around?<br></div><div style="font-family:Arial;">&gt;&nbsp;<br></div><div style="font-family:Arial;">&gt; Thanks,<br></div><div style="font-family:Arial;">&gt; Jon<br></div><div style="font-family:Arial;">&gt;&nbsp;<br></div><div style="font-family:Arial;">&gt;&nbsp;<br></div><div style="font-family:Arial;">&gt;&gt; On Sun, Sep 15, 2019, at 7:55 AM, Andrew Swan wrote:<br></div><div style="font-family:Arial;">&gt;&gt; You might have already seen this, but I have a paper on some related&nbsp;<br></div><div style="font-family:Arial;">&gt;&gt; issues at https://arxiv.org/abs/1808.00920 . In that paper I didn't&nbsp;<br></div><div style="font-family:Arial;">&gt;&gt; look at the original version of regularity, but a more recent version&nbsp;<br></div><div style="font-family:Arial;">&gt;&gt; ("all monomorphisms are cofibrations") that fits better with the&nbsp;<br></div><div style="font-family:Arial;">&gt;&gt; general framework of Orton and Pitts. In that case it is definitely&nbsp;<br></div><div style="font-family:Arial;">&gt;&gt; equality of objects that causes problems.<br></div><div style="font-family:Arial;">&gt;&gt;&nbsp;<br></div><div style="font-family:Arial;">&gt;&gt; Best,<br></div><div style="font-family:Arial;">&gt;&gt; Andrew<br></div><div style="font-family:Arial;">&gt;&gt;&nbsp;<br></div><div style="font-family:Arial;">&gt;&gt;&gt; On Friday, 13 September 2019 08:10:42 UTC+2, Jasper Hugunin wrote:<br></div><div style="font-family:Arial;">&gt;&gt;&gt; Hello all,<br></div><div style="font-family:Arial;">&gt;&gt;&gt;&nbsp;<br></div><div style="font-family:Arial;">&gt;&gt;&gt; I've been trying to understand better why composition for the universe does not satisfy regularity.<br></div><div style="font-family:Arial;">&gt;&gt;&gt; Since comp^i [ phi |-&gt; E ] A is defined as (roughly) Glue [ phi |-&gt; equiv^i E ] A, I would expect regularity to follow from two parts:<br></div><div style="font-family:Arial;">&gt;&gt;&gt; 1. That Glue [ phi |-&gt; equivRefl A ] A reduces to A (a sort of regularity condition for the Glue type constructor itself)<br></div><div style="font-family:Arial;">&gt;&gt;&gt; 2. That equiv^i (refl A) reduces to equivRefl A<br></div><div style="font-family:Arial;">&gt;&gt;&gt; I'm curious as to which (or both) of these parts was the issue, or if regularity for the universe was supposed to follow from a different argument.<br></div><div style="font-family:Arial;">&gt;&gt;&gt;&nbsp;<br></div><div style="font-family:Arial;">&gt;&gt;&gt; Context:<br></div><div style="font-family:Arial;">&gt;&gt;&gt; I've been studying and using CCHM cubical type theory recently, and often finding myself wishing that J computed strictly.<br></div><div style="font-family:Arial;">&gt;&gt;&gt; If I understand correctly, early implementations of ctt did have strict J for Path types, and this was justified by a "regularity" condition on the composition operation, but as discussed in this thread on the HoTT mailing list &lt;https://groups.google.com/d/msg/homotopytypetheory/oXQe5u_Mmtk/3HEDk5g5uq4J&gt;, the definition of composition for the universe was found to not satisfy regularity.<br></div><div style="font-family:Arial;">&gt;&gt;&gt; I don't remember seeing the regularity condition defined anywhere, but my understanding is that it requires that composition in a degenerate line of types, with the system of constraints giving the sides of the box also degenerate in that direction, reduces to just the bottom of the box. This seems to be closed under the usual type formers, plus Glue, but not the universe with computation defined as in the CCHM paper &lt;http://www.cse.chalmers.se/~coquand/cubicaltt.pdf&gt; (for trivial reasons and non-trivial reasons; it gets stuck at the start with Glue [ phi |-&gt; equiv^i refl ] A not reducing to anything).<br></div><div style="font-family:Arial;">&gt;&gt;&gt;&nbsp;<br></div><div style="font-family:Arial;">&gt;&gt;&gt; Best regards,<br></div><div style="font-family:Arial;">&gt;&gt;&gt; - Jasper Hugunin<br></div><div style="font-family:Arial;">&gt;&gt;&nbsp;<br></div><div style="font-family:Arial;">&gt;&gt; --&nbsp;<br></div><div style="font-family:Arial;">&gt;&gt; You received this message because you are subscribed to the Google&nbsp;<br></div><div style="font-family:Arial;">&gt;&gt; Groups "Homotopy Type Theory" group.<br></div><div style="font-family:Arial;">&gt;&gt; To unsubscribe from this group and stop receiving emails from it, send&nbsp;<br></div><div style="font-family:Arial;">&gt;&gt; an email to HomotopyTypeTheory+unsubscribe@googlegroups.com.<br></div><div style="font-family:Arial;">&gt;&gt; To view this discussion on the web visit&nbsp;<br></div><div style="font-family:Arial;">&gt;&gt; https://groups.google.com/d/msgid/HomotopyTypeTheory/16b3b92f-069b-468c-94f8-f3859e152338%40googlegroups.com &lt;https://groups.google.com/d/msgid/HomotopyTypeTheory/16b3b92f-069b-468c-94f8-f3859e152338%40googlegroups.com?utm_medium=email&amp;utm_source=footer&gt;.<br></div><div style="font-family:Arial;">&gt;&nbsp;<br></div><div style="font-family:Arial;">&gt; --&nbsp;<br></div><div style="font-family:Arial;">&gt; You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group.<br></div><div style="font-family:Arial;">&gt; To unsubscribe from this group and stop receiving emails from it, send an email to HomotopyTypeTheory+unsubscribe@googlegroups.com.<br></div><div style="font-family:Arial;">&gt; To view this discussion on the web visit https://groups.google.com/d/msgid/HomotopyTypeTheory/80a2be3b-afaa-4106-9c69-0df6567ec709%40www.fastmail.com.<br></div><div style="font-family:Arial;"><br></div><div style="font-family:Arial;"><br></div></blockquote></body></html>

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      parent reply index

Thread overview: 13+ messages / expand[flat|nested]  mbox.gz  Atom feed  top
2019-09-13  6:10 [HoTT] " Jasper Hugunin
2019-09-15  5:57 ` [HoTT] " Jasper Hugunin
     [not found]   ` <CAMWCppk4PWzfZ1HKNLMdAZ=RBC-ARxtJXR8okvwO3raea5gC8Q@mail.gmail.com>
     [not found]     ` <CAGTS-a-SvWWF+br6sKxGj6ufVY=4m830FH9BDg06QJR1vbNFsw@mail.gmail.com>
2019-09-16  2:18       ` Fwd: " Jasper Hugunin
2019-09-16 16:18         ` [HoTT] " Licata, Dan
2019-09-16 17:09           ` Jasper Hugunin
2019-09-16 19:01             ` Licata, Dan
2019-09-16 20:17               ` Jasper Hugunin
2019-09-18 12:16                 ` Anders Mortberg
2019-09-18 12:52                   ` Thierry Coquand
2019-09-15 11:55 ` [HoTT] " Andrew Swan
2019-09-15 22:38   ` Jasper Hugunin
2019-09-16  1:04   ` Jon Sterling
     [not found]     ` <A605E6EE-0101-4390-B50D-A6AEB36FDCC2@icloud.com>
2019-09-16  1:44       ` Jon Sterling [this message]

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